These are some mathematics investigations I have pursued over the years. They may be of some interest to teachers, students, or hobbyists. I try to convey a conceptual understanding, usually without rigorous proof. Some of the lessons are accompanied by questions and suggestions for extensions.

Textil y sintético

Importado

Caucho único

-Water-listo sintético y poliéster la parte superior de las correas es extremadamente durable y se seca rápidamente después de mojarse

-A Amortiguación amortiguada Pad tratado con Microban, un zinc basado en antimicrobiano tecnología que ayuda a prevenir el mal olor AL44 en el talón absorbe el impacto

Tratado con Microban, un antimicrobiano a base de zinc tecnología que ayuda a prevenir el mal olor

-Contoured Soportes de la plantilla de EVA tu arco para mantener tu pie feliz y saludable

Fácil cierre de gancho y lazo entra y sale rápidamente y consigue el ajuste perfecto

Some of the lessons are accompanied by questions and suggestions for extensions.

In my pursuit of a teaching career, I was told to justify the study of each concept by establishing its relevance to the students’ lives. Sorry, but I still have trouble buying into that one. Fortunately, poetry and music are rarely put to that same test. As I glance at the list below, I must concede that it would be difficult to convert any of the lessons into food, shelter, or money. These are things that interested me, and now I understand them better. If there is a reward, it is the fact that every time I do this I get better at figuring things out.

I occasionally get email from people seeking permission to use my work. Feel free to cite, quote, edit, reproduce, or distribute anything on this site. Do not worry about putting my name on it, but please do not put your own name on it.

Many of these presentations use the Geometer’s Sketchpad, but only one of them actually requires it. If you do not have the software, let me suggest that you follow this link to Key Curriculum Press and look into it.

Paul Kunkel

March 16, 2016

Updated the animations on Hanging With Galileo.

October 23, 2015

Updated the animations on Reuleaux Triangle.

October 9, 2015

Updated the animations on the Vernier Scale page of the Geometry of Surveying.

October 8, 2015

Sandalia Sandalia deportiva para Tirra para mujer Teva Buena Buena Powder Blue c0445feUpdated the animations on Linkages.

October 4, 2015

Updated the animations on On the Square.

September 25, 2015

Updated the animations on Tangent Circles.

September 23, 2015

Updated the lead page of Conic Sections and Construction.

The page it replaced had java animations that were not being treated well by modern browsers. New images were generated with the developmental Web Sketchpad, which is producing very satisfying results. This appears to be the future, so expect more of my pages to be similarly updated soon.

February 26, 2015

Added Book VII to the Conics of Apollonius.

This is the final surviving book of the treatise.

October 22, 2014

Added Book VI to the Conics of Apollonius.

June 10, 2014

Added Book V to the Conics of Apollonius.

Includes GSP-3 files

Includes GSP-4 files

Includes GSP-5 files
Contents:

We are given two fixed points in a vertical plane. A particle starts from rest at one of the points and travels to the other under its own weight. Find the path that the particle must follow in order to reach its destination in the briefest time.

This is a famous problem in the calculus of variations. Nothing new is presented here, but the explanation is more thorough than most.

Buffon’s Needle

An old probability exercise is aided by an interactive Sketchpad file. By manipulating the sketch, an intuitive understanding may be gained, even if the student has had no introduction to calculus.

Sandalia Sandalia deportiva para Tirra para mujer Teva Buena Buena Powder Blue c0445fe

Chinese Handcuffs This has no lesson, only questions. It is just an interesting animated file, which invites student investigation. It has applications for geometry,

KEEN sandalia de Venecia para mujer Tierra oscura / Mar Caribe , Sandalias Newport H2 de KEEN para mujer Imán / Coral caliente , Encaje de quimera de Columbia para mujer Rsealev OluKai Ohana Ciruela / negro , Huracán XLT Mujeres Teva - Avellana / Negro Hazel / Black , Sandalias KEEN Venice II-W para mujer Aceite de visón / Tierra oscura , Sandalia deportiva Nolita para mujer Anne Klein Tela de peltre Sandalia Uneek O2-w de mujer KEEN Estate Blue / Harvest Gold , Sandalia Juniper Mujer Chaco Nutria Sandalias KEEN Venice II H2-W para mujer Cornstalk / Safari , Chaco Mujer Zcloud Sport Sandal Tri Opal Sandalia de cuña Jubilee Propet para mujer Corteza múltiple Sandalia Drew Angela para mujer Negro Sandalia de mujer KEEN Rose Brindle / Shitake Chaco Women s Zx2 Classic Athletic Sandal Wink Blue Sandalias KEEN Venice II H2-W para mujer Cornstalk / Safari Sandalias KEEN Mujer Whisper-w Ágata gris / azul ópalo Sandalias Bali KEEN para mujer Beso de uva / Hierba Lavendar Chaco Women s Z2 Classic Athletic Sandal Elm Aqua , Chaco Mujer Zcloud X Sandalia Deportiva Dahlia Eclipse Chaco Mujer Zcloud X Sandalia Deportiva Durazno estándar Sandalia KEEN Women s Exo-W Uneek Etherea / Plaza Taupe Sandalia NIKE Kawa Slide para mujer Blanco / rosa fuego / cometa azul / aluminio Sandalia UEek Flat-W de mujer KEEN Cuervo azul , Chaco Mujer Zcloud Sport Sandal Agrupado azul , MBT Sandalia 400352-03 KISUMU3 Negro Negro , Sandalia Deportiva Jordan Spires de Via Spiga Albaricoque gamuza , KEEN Hombres Newport H2 Sandal Cuervo / Aluminio , KEEN Hombres Newport H2 Sandal Imán / Capuchina ,

trigonometry, and probability. Fair warning: The discussion revolves about
a figure that appears only when the sketch is animated. Without Geometer’s
Sketchpad, this one will make little sense at all.
Conic Sections and Construction

My emphasis is on geometric properties of the conic sections, particulary understanding that has been handed down to us in ancient texts. There also are some challenging constructions.

The Geometry of Surveying

This used to be my job, and I enjoyed studying all of the geometrical concepts that were part of the science of land surveying. In this article I discuss the instruments of the profession, their operating principles, their error, and their calibration.

Hanging with Galileo

An investigation into the catenary, the curve formed by a chain suspended at both ends. It also extends into the curve of the main cables on a suspension bridge, which, interestingly, is not a catenary.

Inversion Geometry Sandalias OLKAKAI Ohana - Hombre Sombra negra / oscura Chaco Mujer ZX1 Classic Sport Sandalia Paloma mandarina , Zapatillas de running ASICS Gel-Nimbus 18 para mujer Blanco / amarillo de seguridad / acuario Zapatillas NIKE Flex Adapt TR Cross Trainer para mujer Teal Green Zapato deportivo ASICS Gel-Excite 4 para mujer Shark / Flash Coral / Parachute Purple Short NIKE Tempo para mujer Negro Negro , Zapatillas de running Pureboost X de adidas Performance para mujer Ray Red / Vapor rosa / blanco , Zapatillas de running NIKE para mujer Air Presto Gris Salomon Women s Speedcross 4 Rosa virtual Vionic Rest Farra - Sandalias de apoyo para mujeres Corcho de oro , Klogs Footwear Salem Clog para mujer Tawney , Vionic Rest Farra - Sandalias de apoyo para mujeres Corcho de oro , Zapatillas de running Nike Kids Air Max Zero Essential GS Negro / Negro-supremo azul Zapato para caminar Ryka Women s Sky Negro / rosa , J Slides Women s Ellie Sandal Broncearse Birkenstock Blue Footbed-Arch Soporte plano Caramelo New Balance 711v3 Cush + Cross Trainer para mujeres Blanco K-Swiss St329 CMF Sneaker para mujer Angel Wing / Blanco , Sandalia Kinsley de Naturalizer para mujer Negro Merrell Ceylon Sport Ballet Casual Slip On para mujer Castillo de roca , Vans Old Skool Unisex Adults Low-Top Trainers Armada Mujer NIKE SF AIR FORCE 1 MID Vast Grey / Vast Grey , Zapatillas de golf adidas Climacool Knit Cblack / D, mujer, de adidas Luz Onix Sandalia Beau Slide Kate Spade New York para mujer Oro VANS Unisex Authentic Black Canvas VN000EE3BLK Skate Shoe True White Azul de Betsey Johnson Zapatillas SB-Haven para mujer Sonrojo , Bota de petróleo Aerosoles para mujer Rojo oscuro Bota omnidireccional Columbialy Shortly Shorty para mujer Sal marina / cervatillo CLARKS Raisie Vita Boot para mujer Cuero de oliva , Tiempo libre para mujeres de SAS Moca ,

If you have never studied inversion geometry, you can start here. The elementary concepts are explained. This page is a quick reference for the other lessons that employ inversion geometry.

Linkages

Draw a line segment. Did you use a straightedge? How do you know it was straight?

Java simulations are used to investigate a nineteenth century mechanics problem.

On the Square

The challenge: Given four arbitrary, unique points, *A*, *B*,
*C*, and *D*, construct one line through each point such that the four lines form a square.

Orthocenter What is so significant about the orthocenter of a triangle? Why does it even have a name? This investigation was prepared for a geometry teachers’ workshop. It includes a lesson handout in Word format. This lesson was written for use in a computer lab with the Geometer’s Sketchpad, but with a little editing it should work just as well with Cabri or Cinderella.

The Planimeter

A planimeter is a drafting instrument used to measure area. Its mechanism is quite simple, but understanding why it works is more elusive.

Platonic Solids

Find the inradius, circumradius, dihedral angle, surface area, and volume of all of the Platonic solids. There is also some information about some of their other interesting properties.

Reuleaux Triangle

It is not a circle, but it has constant width. Study the properties of this figure with interactive animated sketches.

Rungs and Vineyards

How is that for a descriptive title? Here are investigations of two interesting visual effects seen from a moving automobile. It includes two interactive perspective drawings, and a discussion of the cylindrical and planar projections used to create them.

The Sliding Triangle

An interesting accidental discovery – Investigate ellipses by tracing triangle transformations.

Table Patterns

Explore geometric patterns formed in a table of modular multiplication. This one requires some elementary knowledge of congruence classes and modular arithmetic.

Tangent Circles

Three circles are given. How many circles can be constructed tangent to all three? How are they constructed? This lesson includes ten animations covering all of the special cases. Some of them are very complex. A little knowledge of circle inversions would help.

Volume of a Torus Do you already know the volume of a torus? Then humor me. I just figured Sandalia Sandalia deportiva para Tirra para mujer Teva Buena Buena Powder Blue c0445fe it out, and I got a chance to try out this new graphics software.

Resources

The Geometer’s Sketchpad Workshop

It is plain to see that I am very fond of Sketchpad. This page was created after the release of version 4. The idea is to have a place to exchange ideas with other users, especially those who are using the advanced features. Here are some custom tools and some information on Java Sketchpad.

Sketchpad Gallery These are some of my best sketches. You may find them useful or interesting by themselves.

Geometry Construction Reference

Thirteen elementary straightedge
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and compass constructions are described and illustrated. The original version,
in Word format, can be downloaded and distributed.

Last update: March 16, 2016 ... Paul Kunkel

For email to reach me, the word